1 Oklahoma State University Spears School of Business Time Value of Money
2 Slide 2 Time Value of Money Which would you rather receive as a sign-in bonus for your new job? 1. $15,000 cash upon signing the contract 2. $15,000 cash at the end of first year of employment 3. $20,000 cash at the end of the first of employment Cash in hand seems to be no brainer, but! How would be determine whether (3) is better or worse than (1)?
3 Slide 3 The One-Period Case If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500. $500 would be interest ($10,000.05) $10,000 is the principal repayment ($10,000 1) $10,500 is the total due. It can be calculated as: $10,500 $10,000 (1.05) The total amount due at the end of the investment is call the Future Value(FV FV).
4 Slide 4 Future Value In the one-period case, the formula for FVcan be written as: FV C 0 (1 + r) Where C 0 is cash flow today (time zero), and r is the appropriate interest rate.
5 Slide 5 Present Value If you were to be promised $10,000 due in one year when interest rates are 5-percent, your investment would be worth $9, in today s dollars. $ 9, The amount that a borrower would need to set aside today to be able to meet the promised payment of $10,000 in one year is called the Present Value(PV PV). Note that $10,000 $9, (1.05). $10,
6 Slide 6 Present Value In the one-period case, the formula for PVcan be written as: PV C r Where C1 is cash flow at date 1, and r is the appropriate interest rate.
7 Slide 7 Net Present Value The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment. Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?
8 Slide 8 Net Present Value PV PV $10,000 $9, $9,500+ $9, PV $23.81 The present value of the cash inflow is greater than the cost. In other words, the Net Present Value is positive, so the investment should be purchased.
9 Slide 9 Net Present Value In the one-period case, the formula for NPVcan be written as: NPV Cost+ PV If we had notundertaken the positive NPVproject considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FVwould be less than the $10,000 the investment promised, and we would be worse off in FVterms : $9,500 (1.05) $9,975 < $10,000
10 Slide 10 The Multiperiod Case The general formula for the future value of an investment over many periods can be written as: FV C 0 (1 + r) T Where C 0 is cash flow at date zero, r is the appropriate interest rate, and Tis the number of periods over which the cash is invested.
11 Slide 11 The Multiperiod Case, Cont d The general formula for the present value of an investment over many periods can be written as: PV Where C T is cash flow at date T, C 1 T ( r) T r is the appropriate interest rate, and Tis the number of periods over which the cash is invested.
12 Slide 12 Future Value Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years. What will the dividend be in five years? FV C 0 (1 + r) T $5.92 $1.10 (1.40) 5
13 Slide 13 Future Value and Compounding Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend: $5.92 > $ [$ ] $3.30 This is due to compounding.
14 Slide 14 Future Value and Compounding 3 $ 1.10 (1.40) 2 $ 1.10 (1.40) $ 1.10 (1.40) 5 $ 1.10 (1.40) 4 $ 1.10 (1.40) $1.10 $1.54 $2.16 $3.02 $4.23 $
15 Slide 15 Present Value and Discounting How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%? PV $20, $ 9, $20,000 5 (1.15)
16 Slide 16 How Long is the Wait? If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000? FV C (1 r) 0 + T T $10,000 ( 1.10) $5,000 T ln( 1.10) $ 10,000 $5,000 ln(2) 2 T (1.10) T ln(2) ln(1.10) years
17 Slide 17 What Rate Is Enough? Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your child s education? T 12 C0 (1+ r $ 50,000 $5,000 (1+ r) FV ) About 21.15%. (1+ r) 12 $50,000 $5, ( 1+ r) r
18 Slide 18 Calculator Keys Texas Instruments BA-II Plus FV future value PV present value I/Y periodic interest rate P/Y must equal 1 for the I/Y to be the periodic rate Interest is entered as a percent, not a decimal N number of periods Remember to clear the registers (CLR TVM) after each problem Other calculators are similar in format
19 Slide 19 Multiple Cash Flows Consider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows? If the issuer offers this investment for $1,500, should you purchase it?
20 Slide 20 Multiple Cash Flows , Present Value < Cost Do Not Purchase
21 Slide 21 Valuing Lumpy Cash Flows First, set your calculator to 1 payment per year. Then, use the cash flow menu: CF0 0 CF3 600 I 12 CF1 200 F3 1 NPV F1 1 CF , CF2 400 F4 1 F2 1
22 Slide 22 Compounding Periods Compounding an investment mtimes a year for T years provides for future value of wealth: FV C 1+ 0 r m m T
23 Slide 23 Compounding Periods For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to 2.12 FV $ $50 (1.06) 6 $70.93
24 Slide 24 Effective Annual Rates of Interest A reasonable question to ask in the above example is what is the effective annualrate of interest on that investment? FV $50 (1+ ) $50 (1.06) $ The Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-ofinvestment wealth after 3 years: $50 (1+ EAR) 3 $70.93
25 Slide 25 Effective Annual Rates of Interest FV $50 (1+ EAR) 3 $70.93 EAR (1+ EAR) $70.93 $ $70.93 $ So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually. 3
26 Slide 26 Effective Annual Rates of Interest Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly. What we have is a loan with a monthly interest rate rate of 1½%. This is equivalent to a loan with an annual interest rate of 19.56%. n m r 1 + m (1.015)
27 Slide 27 EAR on a Financial Calculator Texas Instruments BAII Plus keys: description: [2nd] [ICONV] Opens interest rate conversion menu [ ] [C/Y] 12 [ENTER] Sets 12 payments per year [ ][NOM] 18 [ENTER] Sets 18 APR. [ ] [EFF] [CPT] 19.56
28 Slide 28 Continuous Compounding The general formula for the future value of an investment compounded continuously over many periods can be written as: FV C 0 e rt Where C 0 is cash flow at date 0, r is the stated annual interest rate, Tis the number of years, and eis a transcendental number approximately equal to e x is a key on your calculator.
29 Slide 29 Classical Cash Flow Schemes Perpetuity A constant stream of cash flows that lasts forever Growing perpetuity A stream of cash flows that grows at a constant rate forever Annuity A stream of constant cash flows that lasts for a fixed number of periods Growing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods
30 Slide 30 Perpetuity A constant stream of cash flows that lasts forever C C C PV PV C C C ( 1+ r) (1+ r) (1+ r) 3 C r +L
31 Slide 31 Perpetuity: Example What is the value of a British consol that promises to pay 15 every year for ever? The interest rate is 10-percent PV
32 Slide 32 Growing Perpetuity A growing stream of cash flows that lasts forever C C (1+g) C (1+g) 2 0 PV PV C C (1+ g) C (1+ g) ( 1+ r) (1+ r) (1+ r) 3 C r g 2 +L
33 Slide 33 Growing Perpetuity: Example The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? $1.30 $1.30 (1.05) $1.30 (1.05) PV $ $26.00
34 Slide 34 Annuity A constant stream of cash flows with a fixed maturity C C C C L T PV C C C L 2 3 (1+ r) (1+ r) (1+ r) (1 C + r ) T PV C r 1 1 (1+ r) T
35 Slide 35 Annuity: Example If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans? $400 $400 $400 L $ PV $ / ( ) 36 $12,954.59
36 Slide 36 What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? PV $100 (1.09) $100 (1.09) $100 (1.09) $100 (1.09) $100 (1.09) t t 1 $ $ $ $100 $100 $100 $ $ PV $
37 Slide 37 Growing Annuity A growing stream of cash flows with a fixed maturity C 0 1 PV PV C (1+ r) r C g + 1 C (1+ g) 2 (1+ r) C (1+g) 2 1+ (1+ g r ) T + L+ C (1+g) 2 3 L C (1+ g) T (1+ r) C (1+g) T-1 T T 1
38 Slide 38 Growing Annuity: Example A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%? $20, PV $20, $20,000 (1.03) $20,000 (1.03) 39 L 40 $265,121.57
39 Slide 39 Growing Annuity: Example You are evaluating an income generating property. Net rent is received at the end of each year. The first year's rent is expected to be $8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%? 3 $8,500 (1.07) 2 $8,500 (1.07) $8,500 (1.07) $8,500 $8,500 $ 9,095 $9, $10, (1.07) $11, $34,706.26
40 Slide 40 What Is a Firm Worth? Conceptually, a firm should be worth the present value of the firm s cash flows. The tricky part is determining the size, timing, and risk of those cash flows.